Existence of entire solutions of Monge-Ampère equations with prescribed asymptotic behaviors

2018/09/14 by Jiguang Bao, Jingang Xiong, Bao, Jiguang +3
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1809.05421

Abstract

We prove the existence of entire solutions of the Monge-Ampère equations with prescribed asymptotic behavior at infinity of the plane, which was left by Caffarelli-Li in 2003. The special difficulty of the problem in dimension two is due to the global logarithmic term in the asymptotic expansion of solutions at infinity. Furthermore, we give a PDE proof of the characterization of the space of solutions of the Monge-Ampère equation det ∇² u=1 with k≥ 2 singular points, which was established by Gálvez-Martínez-Mira in 2005. We also obtain the existence in higher dimensional cases with general right hand sides.

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